Today is 110th birthday of great Indian mathematician D. R. Kaprekar
Dattaraya Ramchandra Kaprekar (17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers. For his entire career (1930–1962) he was a schoolteacher at Nasik in Maharashtra. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. Kaprekar was once laughed at by most contemporary Indian mathmaticians for his so-called ‘trivial’ play with numbers. It required G. H. Hardy to recognize Ramanujan while Kaprekar’s recognition came through Martin Gardner (he wrote about Kaprekar in his “Mathematical Games” column in March 1975 issue of “Scientific American”)
Here I would discuss a mathemagical trick re-disvovered by Kaprekar called “Gap Filling Process” (though claimed to be present in vedic mathematics)
Gap filling process
This process is magical one and will make you Mathemagician
Let, stand for repeated times (called Repunit ).
Then, we shall denote for -th power of
can be obtained by remembering the expansion for and inserting in the gaps between digits of expansion of with the numbers and alternately, beginning from left to right. No gap is counted after the unit digit.
Let’s see an example:
Find the value of .
Even my scientific calculator fails to calculate this exact value !
Then the gaps are:
Now fill the blanks alternately with and .